 /d dN ch (dN )dN )/(½N  /d part ch  /d ) )/(½N  /d (dN ch part ch

Hydrodynamics as tool for diagnosing Hydrodynamics as tool for diagnosing

heavy ion collisions

heavy ion collisions - - Ia Ia

URL:

Initial state and fluid dynamics

Initial state and fluid dynamics

L.P. Csernai, V.

L.P. Csernai, V. Magas Magas , D. , D. Strottman Strottman , , B. B. B B ä ä uchle uchle , , Yun Yun Cheng, M. Z Cheng, M. Z é é t t é é nyi nyi , and , and

K. K. Tamosiunas Tamosiunas , E. Molnar, J. , E. Molnar, J. Manninen Manninen

TOPICS:

TOPICS:

- - Flow measurements as diagnostic tools Flow measurements as diagnostic tools

- - 3 stages/modules  3 stages/modules  3 dim. CFD calculations 3 dim. CFD calculations

- - Rapid freeze- Rapid freeze -out & hadronization /w out & hadronization /w n n

-scaling - scaling

- - Comments on Viscous Fluid dynamics Comments on Viscous Fluid dynamics

FLOW

 Is fluid dynamics applicable in relativistic nuclear physics?

 Collective Nuclear Flow: Greiner – Koonin [1973 Balaton]

 Transverse Flow Exp. Proof : [1984 Plastic Ball LBL]

• By now: Mc increases – close to macro continuous matter

• Local equilibrium $ EoS / Phase Transition / QGP

(during the middle part of the reaction, / Initial and final stages are out of equilibrium)

• Many flow-patterns are observed in nuclear collisions

Stopping

P+A: [Csernai, Kapusta, PRD31(1985)2795]:y=2.5

R. Stock [CERN (2000)]

Stopping at SPS / NA49

Stopping at RHIC

At RHIC Dy = 9.8 – 10.7, so y-gap = 4-5 !

At RHIC there is also more stopping than

expected. No sign of gap.

Peter Steinberg

Shapes of dN

/d for different N

 

dN

/d  (d N

/d  )/(½N

) dN

/d 

Data

HIJING

HIJING

(d N

/d  )/(½N

)

Systematic error ±(10%

Systematic error ±(10%--20%)20%) 354 216 102

Mean N_part

% 0-3 15-20 35-40

Data

[QM’2001]

Stopping - RHIC

Local equilibrium  Jüttner distr. (MB)

Stationary solution of the BTE , and generalization of the MB distribution

Lorentz

Transformation

Properties:

Kinetic definition of density, energy, momentum

These definitions are applicable for any, equilibrium or non-eq.

situation!

Normalization of Jüttner distribution

From:

=

Similar expressions occur when we evaluate the EoS,

energy density, e, pressure, P, and entropy density, s.

Local equilibrium - Flow - LR frame

( Landau ) Def: Orthogonal proj. to flow

Then: These definitions are applicable for

any, equilibrium or non-equilibrium

situation!

Local equilibrium

• Large no. of degrees of freedom

• Strong Stopping

• Local equilibration 

• Equation of State (EoS) characterizes the equilibrium properties of matter

• Dynamics is well approximated by fluid dynamics (perfect, viscous, …)

• Model predictions become similar

• Multi Module Modeling

EoS from the local eq. phase space distribution

Eg.: From Jüttner  Ideal gas EoS & 2

law of thermodyn. (!)

Pressure – Soft Point?

LBL, AGS, SPS:

Collective flow – P-x vs. y

Pressure sensitive Directed transverse flow decreases with increasing energy.

[D. Rischke, 95]

[E. Shuryak, 95]

[Holme, et al., 89]

But, does it recover

at higher energies ?

[F. Karsch, PASI 2002]

Boltzmann transport equation î phase-space distribution

Conservation laws:

Conservation laws are valid for any distribution f(x,p), however these are not sufficient to determine f(x,p) !

Boltzmann H-theorem: (i) for any f(x,p) the entropy is increasing,

(ii)  stationary solution, where the entropy is maximal

 local equilibrium and  EoS

+ P = P (e,n) ^Solvable for local equilibrium! (0. CE)

Realativistic fluid dynamics

+ η, κ, ... _Solvable for near local equilibrium too! (1. CE)

Relativistic fluid dynamics, more detailed:

RFD must be used not only for large velocities but for large energies and

temperatures also!

Dominant radiation  RHD/RMHD

Multi Module Modeling Multi Module Modeling

• Initial state - pre-equilibrium: Parton Cascade; Coherent Yang-Mills

• Local Equilibrium  Hydro, EoS

• Final Freeze-out: Kinetic models, measurables

• If QGP  Sudden and simultaneous hadronization and freeze out (indicated by HBT, Strangeness, Entropy puzzle)

Landau (1953), Milekhin (1958), Cooper & Frye (1974)

Initial State

• At low energies stopping in SHOCK or DETONATION waves (supersonic!) of width of 1-3fm (possible in Pb+Pb)

[BEVALAC, GSI, AGS]

• Idealized: as discontinuity across a hyper- surface (or layer) in space time.

• Simple solutions of Rel. Fluid dynamics

• Generalized to other stationary processes:

freeze-out, initial equilibration, phase

trans.

Matching Conditions – for S-T hypersurfaces !

  Conservation laws Conservation laws

  Nondecreasing entropy Nondecreasing entropy

Can be solved easily. Yields, via the “Taub adiabat”

and “Rayleigh line”, the final state behind the hyper-

Fusion core - confining hypersurface

Goal:

Goal:

• • fusion is confined into a stable stationary fusion is confined into a stable stationary central domain, with a well defined surface central domain, with a well defined surface (hyper

(hyper- -surface). surface).

• • this ring should be stable, and should this ring should be stable, and should remain in place

remain in place

• • this ring should be diagnosed and this ring should be diagnosed and controlled

controlled

• • a simple general model description is a simple general model description is desirable.

desirable.

„Fire streak” picture – 3 dim.

Myers, Gosset, Kapusta, Westfall

M1

String rope --- Flux tube --- Coherent YM field

Initial stage: Coherent Yang-Mills model

[Magas, Csernai, Strottman, Pys. Rev. C ‘2001]

Yo – Yo Dynamics

Initial state

3

flow component

Initial state – reaching equilibrium

Initial state by V. Magas, L.P. Csernai and D. Strottman Phys. Rev. C64 (01) 014901

M1

Flow is a diagnostic tool Flow is a

Flow is a diagnostic diagnostic tool tool

Impact Impact par. par.

Transparency Transparency – – string tension string tension

Equilibration Equilibration time time

Consequence:

Consequence:

v v

(y), v (y), v

(y), (y), … …

Why should we measure v_1 ???

Why should we measure v_1 ???

Multi Module Modeling Multi Module Modeling

• Initial state - pre-equilibrium: Parton Cascade; Coherent Yang-Mills [Magas]

• Local Equilibrium  Hydro, EoS

• Final Freeze-out: Kinetic models, measurables

• If QGP  Sudden and simultaneous hadronization and freeze out (indicated by HBT, Strangeness, Entropy puzzle)

Landau (1953), Milekhin (1958), Cooper & Frye (1974)

3 3 - - Dim Hydro for RHIC Dim Hydro for RHIC (PIC) (PIC)

M2

Particle in Cell method.

Particle in Cell method.

Better resolution than the Better resolution than the cell- cell - size would allow! size would allow!

“Marker particles “ Marker particles” ” = = Lagrangian

Lagrangian fluid cells. Large fluid cells. Large number of these.

number of these.

Randomly placed to avoid Randomly placed to avoid

“ “ ringing instabilities ringing instabilities ” ” and and

other grid related instabilities!

other grid related instabilities!

Runs very stable up to very Runs very stable up to very high energies, much beyond high energies, much beyond the principle applicability of the principle applicability of CFD approach.

CFD approach.

Figure: In the PIC method Lagrangian fluid elements, called Markers, move in a decartian coordinate grid. At very high energies, to avoid instabilities arising from the computational grid, marker particles are randomized in our approach. The figure shows Marker particle positions in the central plane of an explosion (z is the beam direction), assuming an initial Landau state [15] with an energy density of 40 GeV/fm3. A total of 1.5 million marker particles are used to describe the three-dimensional nucleus [unpublished].

Figure: Test of maximum baryon number density in the explosive final stage of expanding Quark-gluon Plasma after an ultra-relativistic heavy ion reaction, where the initial collision energy was 65 times the rest mass of the colliding nuclei. The result weakly depends on the ratio of the grid size in the direction of the collision and the length of the time-step which is 0.5 fm/c. Implementation of implicit methods and Newton-Krylov solvers for the relativistic hydrodynamics will significantly decrease the fluctuations and increase the accuracy. (Unpublished.)

Figure: Time evolution of the energy density in the central plane assuming an initial Landau state [15], which can be formed in a central (b=0) collision of two nuclei. The expansion is dominantly in the beam-, z-direction. The dynamics were described by a relativistic three-dimensional hydrodynamic model [unpublished].

AU + AU 65 A

AU + AU 65 A GeVGeV runsruns

dxdx dydy dzdz dtdt T cycT cyc TfinTfin [fm] [fm] [fm] [fm/c] [fm/c]

[fm] [fm] [fm] [fm/c] [fm/c] [fm][fm]

--- --- 425kB b=0.0 _tnc16 .86

425kB b=0.0 _tnc16 .86 .86.86 .172 .038043 9.5107 250 4.172 .038043 9.5107 250 4 1,029kB b=0.0 _tnc24 .575

1,029kB b=0.0 _tnc24 .575 .575.575 .115 .0266 4.154 156 4.0266 4.154 156 4 387kB b=0.1 _tnc16 .86

387kB b=0.1 _tnc16 .86 .86.86 .172 .038043 9.5107 250 4.172 .038043 9.5107 250 4 935kB b=0.1 _tnc24 .575

935kB b=0.1 _tnc24 .575 .575.575 .115 .026630 4.287 161 4.026630 4.287 161 4 322kB b=0.25_tnc16 1.032

322kB b=0.25_tnc16 1.032 1.0321.032 .2064 .038043 9.5107 250 4.2064 .038043 9.5107 250 4 780kB b=0.25_tnc24 .69

780kB b=0.25_tnc24 .69 .69.69 .138 .266 4.900 184 4.138 .266 4.900 184 4 142kB b=0.5 _tnc12 1.832

142kB b=0.5 _tnc12 1.832 1.8321.832 .3664 .038043 9.5107 250 5.3664 .038043 9.5107 250 5 55kB b=0.7 _tnc10

55kB b=0.7 _tnc10 2.200 2.200 2.2002.200 .440 .038043 9.5107 250 5.440 .038043 9.5107 250 5 55kB b=0.7 tnc10_tf6

55kB b=0.7 tnc10_tf6 2.200 2.200 2.2002.200 .440 .038043 9.5107 250 6.440 .038043 9.5107 250 6

num-num-ηη = dx= dx dsds T nT n³³ / dn/ dn²² /c = dx/c = dx * 50 * 200 * 1 =* 50 * 200 * 1 = if

if dxdx = 0.1 -= 0.1 - 1 fm => 1 fm =>

ηη = 100 -= 100 - 1000 1000 MeVMeV / (fm/ (fm²² c)c) [in peripheral collisions up to 2000

[in peripheral collisions up to 2000 MeVMeV / (fm/ (fm²²c) ]c) ] see

see L.P.CsL.P.Cs. . egeg. (9.21). (9.21)

M2

Theoretical [D. Molnar, U. Heinz, et al., ] Theoretical [D. Molnar, U. Heinz, et al., ] ηη= 50 –= 50 –500 MeV/fm2c Re 500 MeV/fm2c Re ºº10 –10 –100100

The numerical

The numerical viscosties viscosties are in the range of the are in the range of the estimated physical viscosity [Csernai

estimated physical viscosity [Csernai- -EtAl EtAl- - JPhys JPhys G31 G 31 (2005) 951 (2005) 951- -57] 57]

(except in

(except in peripehral peripehral reactions where it is larger, due to the reactions where it is larger, due to the considerably larger cell size, but here the applicability of hyd considerably larger cell size, but here the applicability of hydro ro is not the best anyway).

is not the best anyway).

The larger than realistic viscosities in the transverse directio The larger than realistic viscosities in the transverse direction n dissipate more transverse flow to heat, so we get higher T and dissipate more transverse flow to heat, so we get higher T and higher thermal smearing. As our T and thermal distributions higher thermal smearing. As our T and thermal distributions are scalar/isotropic our thermal smearing will be slightly

are scalar/isotropic our thermal smearing will be slightly

over- over -estimated. estimated.

Au+Au

Au+Au at 60+60 A GEV, b= 0.5 (R_pat 60+60 A GEV, b= 0.5 (R_p + R_t+ R_t) at ) at

t=0 t=0

Plotted: E, energy density, [GeV/fm

Plotted: E, energy density, [GeV/fm³³], in the calculational], in the calculational (CM) frame. Contour lines are at(CM) frame. Contour lines are at 50, 100, 150 $[GeV/fm^3]$, and

50, 100, 150 $[GeV/fm^3]$, and E_{maxE_{max} = 190.0 GeV/fm} = 190.0 GeV/fm³³..

Au+Au

Au+Au at 60+60 A GEV, b= 0.5 (R_pat 60+60 A GEV, b= 0.5 (R_p + R_t+ R_t) at ) at

t= 0.571 fm/c t= 0.571 fm/c

Plotted: E, energy density, [GeV/fm³³], in the calculational], in the calculational (CM) frame. (CM) frame.

M2

Au+Au

Au+Au at 60+60 A GEV, b= 0.5 (R_pat 60+60 A GEV, b= 0.5 (R_p + R_t+ R_t) at ) at

t= 1.141 fm/c t= 1.141 fm/c

Plotted: E, energy density, [GeV/fm

Plotted: E, energy density, [GeV/fm³³], in the calculational], in the calculational (CM) frame. Contour lines (CM) frame. Contour lines are at 2, 10, 25, 50 [GeV/fm

are at 2, 10, 25, 50 [GeV/fm³³] and E_{max] and E_{max} = 24.91 GeV/fm} = 24.91 GeV/fm³³..

Au+Au

Au+Au at 60+60 A GEV, b= 0.5 (R_pat 60+60 A GEV, b= 0.5 (R_p + R_t+ R_t) at ) at

t= 1.902 fm/c t= 1.902 fm/c

Plotted: E, energy density, [GeV/fm

Plotted: E, energy density, [GeV/fm³³], in the calculational], in the calculational (CM) frame. Contour (CM) frame. Contour

Au+Au

Au+Au at 60+60 A GEV, b= 0.5 (R_pat 60+60 A GEV, b= 0.5 (R_p + R_t+ R_t) at ) at

t= 3.804 fm/c t= 3.804 fm/c

Plotted: E, energy density, [GeV/fm

Plotted: E, energy density, [GeV/fm³³], in the calculational], in the calculational (CM) frame. Contour lines are at(CM) frame. Contour lines are at 0.25, 1.5, 2, 3 [GeV/fm

0.25, 1.5, 2, 3 [GeV/fm³³] and E_{max] and E_{max} = 3.17 Ge/fm} = 3.17 Ge/fm³³V .V .

Au+Au

Au+Au at 60+60 A GEV, b= 0.25 (R_pat 60+60 A GEV, b= 0.25 (R_p + R_t+ R_t) at ) at

t=0 t=0

Plotted: n, baryon charge density, [1/fm

Plotted: n, baryon charge density, [1/fm³³], in the rest frame of the cell. Contour lines], in the rest frame of the cell. Contour lines

Au+Au

Au+Au at 60+60 A GEV, b= 0.25 (R_pat 60+60 A GEV, b= 0.25 (R_p + R_t+ R_t) at ) at

t = 1.33 fm/c t = 1.33 fm/c

Plotted: n, baryon charge density, [1/fm

Plotted: n, baryon charge density, [1/fm³³], in the rest frame of the cell. Contour ], in the rest frame of the cell. Contour lines are at 0.2, 1.0 , 2.5, 5 times n_0, and

lines are at 0.2, 1.0 , 2.5, 5 times n_0, and N_{maxN_{max} = 6.541 n_0 } = 6.541 n_0

Au+Au

Au+Au at 60+60 A GEV, b= 0.25 (R_pat 60+60 A GEV, b= 0.25 (R_p + R_t+ R_t) at ) at

t = 2.66 fm/c t = 2.66 fm/c

Au+Au

Au+Au at 60+60 A GEV, b= 0.25 (R_pat 60+60 A GEV, b= 0.25 (R_p + R_t+ R_t) at ) at

t = 4.79 fm/c t = 4.79 fm/c

Plotted: n, baryon charge density, [1/fm

Plotted: n, baryon charge density, [1/fm³³], in the rest frame of the cell. Contour ], in the rest frame of the cell. Contour lines are at 0.375, 3.75 times n_0, and

lines are at 0.375, 3.75 times n_0, and N_{maxN_{max} = 4.095 n_0 . } = 4.095 n_0 .

Au + Au Au + Au 65 + 65 65 + 65 A.GeV A.GeV b = 25%

b = 25%

Time = Time = cycles cycles 0.266 fm/c 0.266 fm/c

Energy Energy dens.

dens.

[GeV/fm3]

[GeV/fm3]

Max = Max =

140GeV/fm3 140GeV/fm3

Multi Module Modeling Multi Module Modeling

• Initial state - pre-equilibrium: Parton Cascade; Coherent Yang-Mills [Magas]

• Local Equilibrium  Hydro, EoS

• Final Freeze-out: Kinetic models, measurables

• If QGP  Sudden and simultaneous

hadronization and freeze out (indicated

by HBT, Strangeness, Entropy puzzle)

3-dim Hydro for RHIC Energies

Au+Au E

=65 GeV/nucl. b=0.1 b

A

=0.08 => σ~10 GeV/fm

n / n ₀ [ 1 ] e [ GeV / fm ³ ]

T= 7.6 fm/c n

= 5.22 e

= 37.16 GeV / fm

L

= 1.45 fm L

=0.145 fm

. .

Global Flow

Directed Transverse flow

Elliptic flow

3 ^rd flow

component (anti - flow) 3 ^rd flow

component (anti - flow)

Squeeze out

A= A=

0.065 0.065

11.4 fm/c

3 ^rd flow component

Hydro

[Csernai, HIPAGS’93]

Multi Module Modeling Multi Module Modeling

• Initial state - pre-equilibrium: Parton Cascade; Coherent Yang-Mills [Magas]

• Local Equilibrium  Hydro, EoS

• Final Freeze-out: Kinetic models, measurables

• If QGP  Sudden and simultaneous hadronization and freeze out

(n-q scaling )

Properties of the medium & hadronization mechanisms via identified particle measurements

xy z

 



 



  





1 2

3 3

cos 2

2 1 1

n

n T

T

n dy v

dp p

N d p

d N

E d





Scaling for V ₂

Quark content determines v ₂

From q-recombination models:

v ₂

= v ₂

(p

/ n )/ n

NCQ – NCQ – scaling scaling

analogous analogous to to

N- N - nucleon nucleon scaling!

scaling!

• All particles originate from common flow before

hadronization

•

Scaling is observed over extended range of KE

Kinetic Energy Scaling

Pressure gradients converting work into kinetic energy

KE

 m ( 

 1 )  m

 m

PHENIX, nucl-ex/0608033

What is the scaling variable?

p _T vs. KE _T Scaling

STAR

How is this observation accommodated the recombination approach? > Local energy conservation including flow. =>

KE

/n seems to work better than p

/n with the constituent quark scaling

J. Jia1 and C. Zhang, hep-ph/0608187

Rapid and simultaneous FO and

“hadronization”

• Improved Cooper-Frye FO:

- Conservation Laws:

- Post FO distribution:

• Hadronization ~ CQ-s

- Pre FO: Current and , QGP - Post FO: Constituent and

- are conserved in FO!!!

 ^{  0}   ^,

^

^{ 0} 

 

N T

0 )

( )

(  

 p

f p

q q

q q

q

q

N

N and

Observed

Observed n n

– – scaling scaling

there is no further flow

development after hadronization development after hadronization

R. A. Lacey (2006), nucl-ex/0608046.

b = 70% b = 70%

nn_qq scalingscaling

Transverse momentum distribution of v1, v2, Transverse momentum distribution of v1, v2, n n

Freeze Out:

Freeze Out:

High High pTpT ptcles

ptcles fromfrom outside outside regions FO regions FO and and

hadronize hadronize first!

first!

Rapidity distribution of v

Rapidity distribution of v

, v , v

, nq , nq

b = 70% b = 70%

nn_qq scalingscaling

pp_TT = p= p_TT / / nn_qq

BjöBjörnrn BäBäuchleuchle

FO hypersurface ^T

^{=139 MeV}

[B. Schlei, LANL 2005]

Freeze out:

Freeze out:

V.K. Magas, V.K. Magas, E. Molnar.

E. Molnar.

Improved calculation of FO hypersurface

Flow patterns

• Strong, correlated and dominant “Elliptic”, V ₂ , flow observed (CERN/BNL).

• The flow is laminar (η is sufficiently large), & not dissipated (η is sufficiently small) !?

• V ₁ , „directed flow” measurements are not as detailed yet as V ₂ .

• The strong and dominant flow measurements

raised large, attention!

Viscosity vs. T has a

Viscosity vs. T has a minimum at the 1minimum at the 1^stst order phase transition. order phase transition. This might signal the phase transition if viscosity is measured.

This might signal the phase transition if viscosity is measured.

At lower energies this was done.

At lower energies this was done.

Stability, Reynolds number

- kinematic viscosity

- viscosity - density

- length - velocity

In an ideal fluid any small perturbation increases and leads to turbulent flow. For stability

sufficiently large viscosity and/or heat conductivity are needed!

Re < 1000 - 2000

(Calculations are also stabilized by numerical viscosity!)

Measured

Measured [[LijuanLijuan RuanRuan / STAR]/ STAR]scaling [A, E] of dimensionless vscaling [A, E] of dimensionless v22 fluctuations , can be fluctuations , can be compared to constant

compared to constant ReRe contours. If the two are similar viscous effects are dominant in contours. If the two are similar viscous effects are dominant in these fluctuations, and viscosity (or

these fluctuations, and viscosity (or Re) can be extracted. ) can be extracted.

Re – studies in HICs

Theoretical [D. Molnar, U. Heinz, et al., ] Theoretical [D. Molnar, U. Heinz, et al., ] η η = 50 = 50 – – 500 MeV/fm 500 MeV/fm

c Re c Re º º 10 10 – – 100 100 Exp.: 50

Exp.: 50 – – 800 800 Mev/nucleon energies 80 Mev /nucleon energies 80’ ’s s [Bonasera [ Bonasera , Schurmann , Schurmann , Csernai] , Csernai]

scaling analysis of flow parameters.

scaling analysis of flow parameters. Re Re º º 7 7 – – 8 ! 8 ! (more dilute, more viscous matter)

(more dilute, more viscous matter)

In both cases

In both cases η η/s /s ª ª 1 (0.5 – 1 (0.5 – 5) , 5) ,

This is a value large enough to keep the This is a value large enough to keep the flow laminar in Heavy Ion Collisions !!!

flow laminar in Heavy Ion Collisions !!!

OUTLOOK next:

OUTLOOK next:

- - Flow measurements as diagnostic tools Flow measurements as diagnostic tools – – still can provide a lot of information.

still can provide a lot of information.

- - 3 dim. CFD calculations, are important but 3 dim. CFD calculations, are important but complex task:

complex task: EoS EoS , initial , initial - - and final states. and final states.

- - Viscous Fluid dynamics: Important and can Viscous Fluid dynamics: Important and can

help to locate the phase transition threshold.

help to locate the phase transition threshold.

Numerical viscosity is important.

Numerical viscosity is important.

- - More about fusion plasma is coming ! More about fusion plasma is coming !

End of Part Ia